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$e^{A}$ denotes the exponential of a square matrix $A$. Suppose $\lambda$ is an eigenvalue and $v$ is the corresponding eigen-vector of the matrix $A$.

Consider the following two statements:

Statement $1$ : $e^{\lambda }$ is an eigenvalue of $e^{A}$.

Statement $2$ : $v$ is an eigen-vector of $e^{A}$.

Which one of the following options is correct?

- Statement $1$ is true and statement $2$ is false.
- Statement $1$ is false and statement $2$ is true.
- Both the statements are correct.
- Both the statements are false.